IMO Shortlist 1996 problem C1
Kvaliteta:
Avg: 0,0Težina:
Avg: 6,0 We are given a positive integer
and a rectangular board
with dimensions
. The rectangle is divided into a grid of
unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is
. The task is to find a sequence of moves leading from the square with
as a vertex to the square with
as a vertex.
(a) Show that the task cannot be done if
is divisible by 2 or 3.
(b) Prove that the task is possible when
.
(c) Can the task be done when
?
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![AB = 20, BC = 12](/media/m/d/0/8/d08b29385eb1725ab215df291b892235.png)
![20 \times 12](/media/m/7/0/2/70222e5944b67df00fe34f174ac77a12.png)
![\sqrt {r}](/media/m/7/6/1/76139539508913d2e7025810eca3e232.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
(a) Show that the task cannot be done if
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
(b) Prove that the task is possible when
![r = 73](/media/m/9/3/e/93efed3367342feaa876721c4ec4606f.png)
(c) Can the task be done when
![r = 97](/media/m/d/1/0/d102b3eb86567d1cc0f0e6d9de72239c.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1996